Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
$\mathbb{R}$-dimension: | $4$ |
Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
Name: | $S_4$ |
Order: | $24$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$26$ |
$0$ |
$345$ |
$0$ |
$5824$ |
$0$ |
$116004$ |
$0$ |
$2611356$ |
$a_2$ |
$1$ |
$2$ |
$9$ |
$51$ |
$358$ |
$2902$ |
$26245$ |
$258575$ |
$2727940$ |
$30396804$ |
$353744551$ |
$4260593031$ |
$52727046564$ |
$a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$810$ |
$0$ |
$106345$ |
$0$ |
$20169478$ |
$0$ |
$4778002866$ |
$0$ |
$1291783829820$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$26$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$51$ |
$31$ |
$91$ |
$56$ |
$174$ |
$345$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$71$ |
$358$ |
$216$ |
$134$ |
$689$ |
$420$ |
$1378$ |
$835$ |
$2810$ |
$5824$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$531$ |
$2902$ |
$322$ |
$1735$ |
$1046$ |
$5878$ |
$3514$ |
$2114$ |
$12164$ |
$7245$ |
$25523$ |
$15134$ |
$54152$ |
$116004$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$810$ |
$4473$ |
$26245$ |
$2676$ |
$15482$ |
$9180$ |
$55078$ |
$5463$ |
$32402$ |
$19133$ |
$117150$ |
$68654$ |
$40386$ |
$251669$ |
$$ |
$146944$ |
$545174$ |
$317184$ |
$1189440$ |
$2611356$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&9&0\\1&0&6&0&4&0&9&9&0&4&0&12&0&0&20\\0&2&0&5&0&8&0&0&8&0&3&0&12&14&0\\1&0&4&0&7&0&9&10&0&7&0&16&0&0&24\\0&6&0&8&0&26&0&0&28&0&14&0&36&52&0\\1&0&9&0&9&0&24&21&0&13&0&35&0&0&64\\2&0&9&0&10&0&21&29&0&16&0&37&0&0&72\\0&6&0&8&0&28&0&0&40&0&16&0&46&70&0\\1&0&4&0&7&0&13&16&0&17&0&24&0&0&52\\0&3&0&3&0&14&0&0&16&0&13&0&23&34&0\\1&0&12&0&16&0&35&37&0&24&0&72&0&0&124\\0&5&0&12&0&36&0&0&46&0&23&0&76&98&0\\0&9&0&14&0&52&0&0&70&0&34&0&98&151&0\\2&0&20&0&24&0&64&72&0&52&0&124&0&0&260\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&26&24&29&40&17&13&72&76&151&260&116&123&254&191&56\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.